Oblique shock waves#

Introduction#

This tutorial extends the analysis of normal shock waves to the case of oblique shocks, where the shock front is inclined relative to the incoming supersonic flow. In this configuration, the post-shock state can be determined by prescribing either the shock wave angle \(\beta\) (angle between the upstream flow direction and the shock) or the flow deflection angle \(\theta\) (turning of the flow across the shock), as illustrated below:

../../_images/sketch_oblique_shock.svg

Fig. 7 Schematic of an oblique shock wave with wave angle \(\beta\) and flow deflection angle \(\theta\).#

Governing equations#

The Rankine–Hugoniot jump conditions across a planar, inviscid, adiabatic oblique shock are equivalent to those for a normal shock when written in terms of the normal component of the velocity. Specifically,

\begin{equation} p_2 = p_1 + \rho_1 u_{n,1}^2 \left( 1-\dfrac{\rho_1}{\rho_2}\right) \quad \text{and} \quad h_2 = h_1 + \dfrac{u_{n,1}^2}{2}\left[1- \left(\dfrac{\rho_1}{\rho_2}\right)^2\right], \end{equation}

where \(u_{n,1} = \mathcal{M}_1 a_1 \sin\beta\) is the upstream velocity component normal to the shock, with \(\mathcal{M}_1\) the upstream Mach number and \(a_1\) the upstream speed of sound. Note that the tangential velocity remains continuous across an inviscid shock in the absence of shear stress.

The wave angle \(\beta\) and flow deflection angle \(\theta\) are related through the oblique-shock trigonometric condition derived from tangential momentum conservation (\(u_{1,t} = u_{2,t}\)):

\begin{equation} \tan(\beta - \theta) = \frac{u_{2,n}}{u_{1,n}} \tan\beta \end{equation}

where \(u_{2,n} = u_2 \sin(\beta - \theta)\) is the downstream velocity component normal to the shock, and \(u_{2,t}\) the tangential component.

Solution branches#

If the wave angle \(\beta\) is prescribed, the Rankine–Hugoniot equations yield a unique post-shock solution. However, if the deflection angle \(\theta\) is given, two physically admissible solutions may exist:

  • Weak solution (smaller \(\beta\)): the downstream flow remains supersonic.

  • Strong solution (larger \(\beta\)): the downstream flow becomes subsonic.

These two branches converge at the maximum deflection angle \(\theta_{\max}\), above which no physical solution exists for a given upstream Mach number \(\mathcal{M}_1\). In practice, the weak-shock branch is usually observed in external aerodynamic flows, while the strong-shock branch is less common.

Numerical examples#

As for the normal shock case, we employ the ShockSolver() class, part of the +combustiontoolbox.+shockdetonation (CT-SD) subpackage (module). Below is an example that solves the Rankine-Hugoniot equations for a planar oblique incident shock wave in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume), with an initial temperature \(T_1 = 300\) K, pressure \(p_1 = 1\) bar, pre-shock Mach number \(\mathcal{M}_1 = 10\), and wave angle \(\beta = 20:1:85\) degrees:

% Import packages
import combustiontoolbox.databases.NasaDatabase
import combustiontoolbox.core.*
import combustiontoolbox.shockdetonation.ShockSolver

% Get NASA's database
DB = NasaDatabase();

% Define chemical system
system = ChemicalSystem(DB);

% Initialize mixture
mix = Mixture(system);

% Define chemical state
set(mix, {'N2', 'O2'}, [79/21, 1]);

% Define properties
mixArray1 = setProperties(mix, 'temperature', 300, 'pressure', 1, 'Mach', 10, 'beta', 20:1:85);

% Initialize shock solver
solver = ShockSolver();

% Perform shock calculations
[mixArray1, mixArray2] = solveArray(solver, mixArray1);

% Generate report
report(solver, mixArray1, mixArray2);

This code snippet will generate two figures: Molar fraction of species in the mixture as a function of the wave angle \(\beta\) and the variation of the thermodynamic properties (e.g., temperature, pressure) as a function of \(\beta\).

../../_images/shock_waves_2_fig1_beta.svg

Fig. 8 Molar fraction of chemical species downstream of an oblique shock as a function of the wave angle \(\beta\), for a Mach 10 in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) at standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.#

../../_images/shock_waves_2_fig2_beta.svg

Fig. 9 Thermodynamic properties downstream of an oblique shock in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) as function of the wave angle \(\beta\), for a fixed upstream Mach number \(\mathcal{M}_1 = 10\), and standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.#

As for the normal shock case, we employ the ShockSolver() class, part of the +combustiontoolbox.+shockdetonation (CT-SD) subpackage (module). Below is an example that solves the Rankine-Hugoniot equations for a planar oblique incident shock wave in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume), with an initial temperature \(T_1 = 300\) K, pressure \(p_1 = 1\) bar, pre-shock Mach number \(\mathcal{M}_1 = 10\), and theta angle \(\theta = 5:1:50\) degrees:

% Import packages
import combustiontoolbox.databases.NasaDatabase
import combustiontoolbox.core.*
import combustiontoolbox.shockdetonation.ShockSolver

% Get NASA's database
DB = NasaDatabase();

% Define chemical system
system = ChemicalSystem(DB);

% Initialize mixture
mix = Mixture(system);

% Define chemical state
set(mix, {'N2', 'O2'}, [79/21, 1]);

% Define properties
mixArray1 = setProperties(mix, 'temperature', 300, 'pressure', 1, 'Mach', 10, 'theta', 5:1:50);

% Initialize shock solver
solver = ShockSolver();

% Perform shock calculations
[mixArray1, mixArray2_1, mixArray2_2] = solveArray(solver, mixArray1);

% Generate report
report(solver, mixArray1, mixArray2_1, mixArray2_2);

This code snippet will generate three figures: Molar fraction of species in the mixture as a function of the deflection angle \(\theta\) for the weak and strong solutions, as well as the variation of the thermodynamic properties (e.g., temperature, pressure) as a function of \(\theta\) for both branches.

../../_images/shock_waves_2_fig1_theta_weak.svg

Fig. 10 Molar fraction of chemical species downstream of the weak oblique shock as a function of the flow deflection angle \(\theta\), for a Mach 10 in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) at standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.#

../../_images/shock_waves_2_fig1_theta_strong.svg

Fig. 11 Molar fraction of chemical species downstream of the strong oblique shock as a function of the flow deflection angle \(\theta\), for a Mach 10 in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) at standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar.#

../../_images/shock_waves_2_fig2_theta.svg

Fig. 12 Thermodynamic properties downstream of an oblique shock in air (79% \(\text{N}_2\), 21% \(\text{O}_2\) by volume) as function of the flow deflection angle \(\theta\), for a fixed upstream Mach number \(\mathcal{M}_1 = 10\), and standard upstream conditions \(T_1 = 300\) K, \(p_1 = 1\) bar. The blue line corresponds to the weak solution, while the red line corresponds to the strong solution.#